symbolic computation - Need help understanding finite fields / modulo for polynomials

I'm taking a class in finite fields and have not been able to conceptualize how modulo + finite fields works in polynomial space. I understand the basic premises of modular arithmetic, but can't work out how to actually generate a finite field of polynomials.

For example:

Find all $f(x)$ and $g(x)$ in $\mathbb Z_3[x]$:
$$(x^3 + x +1) f(x) + (x^2 + x +1)g(x) = 1$$

I know conceptually how to solve this sort of equation when the coefficients are integers and $f(x), g(x)$ are simple variables, but I don't know how to generate fields in $\mathbb Z_3[x]$ and then how exactly to use them to solve this sort of equation for polynomials once I have their $\mathrm{gcd}$ in $\mathbb Z_3[x]$.